That would presumably raise rents in surrounding counties, which might actually make things worse. A big hunk of the homeless come from other counties, and went to SF after the fact:
"Ten percent (10%) of respondents reported they were living out of state at the time they lost their housing.
Nineteen percent (19%) reported they were living in another county in California. Six percent (6%) reported
they were living in Alameda County at the time, 3% San Mateo, 2% Marin, 2% Contra Costa and 1% Santa Clara
County."
"Ten percent (10%) of respondents reported they were living out of state at the time they lost their housing. Nineteen percent (19%) reported they were living in another county in California. Six percent (6%) reported they were living in Alameda County at the time, 3% San Mateo, 2% Marin, 2% Contra Costa and 1% Santa Clara County."
https://sfgov.org/lhcb/sites/default/files/2015%20San%20Fran...